Question

Two trains leave the railroad station at noon. The first train travels along a straight track at 95 mph. The second train travels at 75 mph along another straight track that makes an angle of 130° with the first track. At what time are the trains 400 miles apart? Round your answer to the nearest minute.

Answer 1

2:36pm

Explanation:

Let's say it takes x hours after leaving the station when the trains are 400 miles apart

Let's refer to the two trains using the letters A and B for easier reference

A is the train that travels on a straight track due East
B is the train that travels on a track 130° relative to the path of train A

A travels at a speed of 95 mph
B travels at a speed of 75 mph

Let the distance between the two trains be 400 miles after they have traveled an unknown
`x` hours. We have to determine
`x`

After
`x` hours, train A would be
`95x` miles from the railroad station and train B would be
`75x` miles from the station

We can visualize this situation as given in the attached image.

You can see that A = 95x, B = 75x and the longest side is 400 miles

We will use the law of cosines to determine x

The Law of Cosines (also called the Cosine Rule) says:

`c^2 = a^2 + b^2 - 2ab\cos (\theta)`

where
`\theta` is the angle opposite c i.e. the angle bounded by a and b which are the other two sides

Using the diagram as a reference we see

c = 400, a = 95x, b= 75x,
`\theta =` 130°

Using this information and plugging values into the cosine rule formula we get

`c^2 = a^2 + b^2 -2ab \cos(\theta)\\\\c^2 = (95x)^2 + (75x)^2 - 2(95x)(75x)\cos(130^\circ)\\\\c^2 = 9025x^2 + 5625x ^2 - 14250x^2\cdot \cos(130^\circ)\\\\`

`c^2 = x^2(9025 + 5625 -14250 \cdot \cos130^\circ)\\\\\\c^2 = x^2(14650 - (14250 * -0.64278))\\`

`c^2 = (14650 - (-9159.72343))x^2\\\\c^2 = (14650 + 9159.72343)x^2 \\\\c^2 \approx 23809.7234x^2\\\\`

But we know that c = 400 since that's what the question says
So we get

`23809.7234x^2 = 400^2\\\\23809.7234x^2 = 160000\\\\x^2 = (160000)/(23809.7234) \approx 6.71994\\\\\\x = √(6.71994) = 2.59228\\\\\textrm{Rounding to nearest tenth}\\x = 2.6 \textrm{ hours}\\\\`

So the two trains will be 400 miles apart 2.6 hours after they have left the station.

2.6 hours = 2 hours and 36 minutes

So the trains will be 400 miles 2 hours and 36 minutes after they have left the station

Since they have left at noon, they are 400 miles apart at 2:36pm